Large Degree Asymptotics of Generalized Bernoulli and Euler Polynomials

Details Large Degree Asymptotics of Generalized Bernoulli and Euler Polynomials Large Degree Asymptotics of Generalized Bernoulli and Euler Polynomials

2009-09-17

arxiv.org/abs/0909.3184v1

Mathematics

Classical Analysis and ODEs

20 pages, 1 figure

Scientific paper

10.1016/j.jmaa.2009.08.034

Asymptotic expansions are given for large values of $n$ of the generalized Bernoulli polynomials $B_n^\mu(z)$ and Euler polynomials $E_n^\mu(z)$. In a previous paper L\'opez and Temme (1999) these polynomials have been considered for large values of $\mu$, with $n$ fixed. In the literature no complete description of the large $n$ asymptotics of the considered polynomials is available. We give the general expansions, summarize known results of special cases and give more details about these results. We use two-point Taylor expansions for obtaining new type of expansions. The analysis is based on contour integrals that follow from the generating functions of the polynomials.

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